∫ 2 0 x ( 2 + x 5 ) d x = ∫ 2 0 ( 2 x + x 6 ) d x = − ∫ 0 2 ( 2 x + x 6 ) d x = ( 2 x 2 1 + 1 + x 6 + 1 6 + 1 ) | 0 2 = ( x 2 + x 7 7 ) | 0 2 = ( ( 2 ) 2 + ( 2 ) 7 7 ) − ( ( 0 ) 2 + 0 7 7 ) = 4 + 2 7 7 = 156 7 {\displaystyle {\begin{aligned}\int _{2}^{0}x(2+x^{5})\,dx=\int _{2}^{0}(2x+x^{6})\,dx&=-\int _{0}^{2}(2x+x^{6})\,dx=\left({\frac {2x^{2}}{1+1}}+{\frac {x^{6}+1}{6+1}}\right){\bigg |}_{0}^{2}=\left(x^{2}+{\frac {x^{7}}{7}}\right){\bigg |}_{0}^{2}=\left((2)^{2}+{\frac {(2)^{7}}{7}}\right)-\left((0)^{2}+{\frac {0^{7}}{7}}\right)=4+{\frac {2^{7}}{7}}={\frac {156}{7}}\end{aligned}}}
